Subgroups — C₃×#13268 #10631875

Order 768.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₃×#2150	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₃×#1377	C₂	1	4	yes	maximal
5	#13268	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₃×M₃₂	C₂²	1	2	yes	—
9	C₃×#446	—	2	1	no	—
10	C₃×#446	—	2	1	no	—
11	C₃×#446	—	2	1	no	—
12	C₃×#446	—	2	1	no	—
13	C₂×C₃×C₄⋊C₈	C₄	1	2	yes	—
14	C₂×C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₃×#97	C₂²	1	2	yes	—
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊C₁₆	—	2	1	no	—
18	C₃×C₄⋊C₁₆	—	2	1	no	—
19	C₃×#346	—	2	1	no	—
20	C₃×C₄×C₁₆	—	2	1	no	—
21	C₂×#446	C₂×C₃	1	1	yes	characteristic
22	#2150	C₂×C₃	1	1	yes	characteristic
23	C₄×M₃₂	C₂×C₃	1	1	yes	characteristic
24	#1377	C₂×C₃	1	4	yes	—
25	C₂²×C₃×Q₈	C₈	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
29	C₃×C₄×Q₈	—	2	1	no	—
30	C₃×C₄×Q₈	—	2	1	no	—
31	C₃×C₄×Q₈	—	2	1	no	—
32	C₃×C₄×Q₈	—	2	1	no	—
33	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₃×C₄⋊C₄	C₈	1	2	yes	—
35	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₃×M₃₂	—	2	2	no	—
38	C₃×M₃₂	—	2	4	no	—
39	C₃×M₃₂	Q₈	1	4	yes	complemented
40	C₂×C₃×C₁₆	—	2	2	no	—
41	C₂×C₃×C₁₆	—	2	2	no	—
42	C₃×C₄⋊C₈	—	2	2	no	—
43	C₃×C₄⋊C₈	—	2	2	no	—
44	C₃×C₄⋊C₈	—	2	1	no	—
45	C₃×C₄⋊C₈	—	2	1	no	—
46	C₃×#97	—	2	2	no	—
47	C₃×#97	—	2	2	no	—
48	C₃×C₄×C₈	D₄	1	1	yes	characteristic
49	C₃×C₄×C₈	D₄	1	1	yes	characteristic
50	C₃×C₄×C₈	D₄	1	1	yes	characteristic
51	C₃×C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
53	C₂×M₃₂	C₂²×C₃	1	2	yes	—
54	C₂×M₃₂	C₂²×C₃	1	2	yes	—
55	#446	—	2	1	no	—
56	#446	—	2	1	no	—
57	#446	—	2	1	no	—
58	#446	—	2	1	no	—
59	C₂×C₄⋊C₈	C₃×C₄	1	2	yes	—
60	C₂×C₄⋊C₈	C₃×C₄	1	1	yes	characteristic
61	C₂×#97	C₂²×C₃	1	2	yes	—
62	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₄⋊C₁₆	—	2	1	no	—
64	C₄⋊C₁₆	—	2	1	no	—
65	#346	—	2	1	no	—
66	C₄×C₁₆	—	2	1	no	—
67	C₂×C₃×Q₈	—	2	1	no	—
68	C₂×C₃×Q₈	—	2	1	no	—
69	C₂×C₃×Q₈	—	4	2	no	—
70	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
72	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
73	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	—	2	2	no	—
77	C₂×C₃×C₈	—	2	4	no	—
78	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
83	C₂×C₃×C₈	#41	1	2	yes	—
84	C₃×C₄⋊C₄	—	2	1	no	—
85	C₃×C₄⋊C₄	—	2	1	no	—
86	C₃×C₄⋊C₄	—	2	2	no	—
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₄²	—	2	2	no	—
89	C₃×C₄²	—	2	2	no	—
90	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
91	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
92	C₃×C₄²	M₁₆	1	1	yes	characteristic
93	C₃×C₄²	M₁₆	1	1	yes	characteristic
94	C₃×C₁₆	—	2	4	no	—
95	C₃×C₁₆	—	4	2	no	—
96	C₂²×Q₈	C₃×C₈	1	1	yes	characteristic
97	C₂²×C₈	C₂×C₃×C₄	1	2	yes	—
98	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
100	C₄×Q₈	—	2	1	no	—
101	C₄×Q₈	—	2	1	no	—
102	C₄×Q₈	—	2	1	no	—
103	C₄×Q₈	—	2	1	no	—
104	C₂×C₄⋊C₄	C₃×C₈	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₃×C₈	1	2	yes	—
106	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
107	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	2	4	no	—
110	M₃₂	C₃×Q₈	1	4	yes	complemented
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	1	no	—
116	C₄⋊C₈	—	2	1	no	—
117	#97	—	2	2	no	—
118	#97	—	2	2	no	—
119	C₄×C₈	C₃×D₄	1	1	yes	characteristic
120	C₄×C₈	C₃×D₄	1	1	yes	characteristic
121	C₄×C₈	C₃×D₄	1	1	yes	characteristic
122	C₄×C₈	C₃×D₄	1	1	yes	characteristic
123	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
124	C₃×Q₈	—	4	2	no	—
125	C₃×Q₈	—	4	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	4	1	no	—
130	C₂×C₃×C₄	—	4	2	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
138	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₃×C₄	#131	1	2	yes	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	—	4	2	no	—
143	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
144	C₃×C₈	#124	1	1	yes	characteristic
145	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
146	C₂×Q₈	—	2	1	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂²×C₄	C₂×C₃×C₈	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₃×C₈	1	2	yes	—
151	C₂²×C₄	C₂×C₃×C₈	1	2	yes	—
152	C₂²×C₄	C₂×C₃×C₈	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
162	C₂×C₈	C₃×#41	1	2	yes	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	C₃×C₂²⋊C₄	1	1	yes	characteristic
170	C₄²	C₃×C₂²⋊C₄	1	1	yes	characteristic
171	C₄²	C₃×M₁₆	1	1	yes	characteristic
172	C₄²	C₃×M₁₆	1	1	yes	characteristic
173	C₁₆	—	2	4	no	—
174	C₁₆	—	4	2	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
178	C₂²×C₃	C₈×Q₈	1	2	yes	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	—	4	2	no	—
184	C₃×C₄	#349	1	2	yes	—
185	C₃×C₄	#391	1	1	yes	characteristic
186	C₃×C₄	#408	1	1	yes	characteristic
187	C₃×C₄	#446	1	2	yes	—
188	C₂³	C₂²×C₃×C₈	1	1	yes	characteristic
189	Q₈	—	4	2	no	—
190	Q₈	—	4	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
199	C₂×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₃×C₂²⋊C₈	1	1	yes	characteristic
201	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₄	C₃×C₄×Q₈	1	2	yes	—
203	C₂×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
204	C₂×C₄	C₃×#131	1	2	yes	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	4	2	no	—
208	C₈	C₃×C₂²⋊Q₈	1	2	yes	—
209	C₈	C₃×#124	1	1	yes	characteristic
210	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	#1984	1	1	yes	characteristic
213	C₂×C₃	C₂×#349	1	1	yes	characteristic
214	C₂×C₃	#2115	1	1	yes	characteristic
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
218	C₂²	C₃×C₈×Q₈	1	2	yes	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	C₃×#349	1	2	yes	—
225	C₄	C₃×#391	1	1	yes	characteristic
226	C₄	C₃×#408	1	1	yes	characteristic
227	C₄	C₃×#446	1	2	yes	—
228	C₃	#13268	1	1	yes	characteristic, direct product factor, Sylow
229	C₂	—	2	2	no	—
230	C₂	C₃×#1984	1	1	yes	characteristic
231	C₂	C₂×C₃×#349	1	1	yes	characteristic
232	C₂	C₃×#2115	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₃×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₃	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. #13268	C₃	direct (N × H)
39. C₃×M₃₂	Q₈	semidirect
110. M₃₂	C₃×Q₈	semidirect
228. C₃	#13268	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#446	2
C₃×#2150	2
C₃×C₄×M₃₂	2
C₃×#1377	2
#13268	3

Subgroups — C3×#13268 #10631875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13268 #10631875